1.2: Extend Numerical Patterns
Remember the trees in the last Concept? Well, Kelly wrote down a pattern of numbers and in the last Concept you figured out a rule for the pattern. Let's look at the pattern again and the rule for the pattern.
1, 1, 2, 3, 5, 8, 13....
This pattern has a rule. This rule is that the two previous numbers add together to equal the next number. Given this information, what is the next number in the pattern?
When you extend number patterns, you can use the rule to figure out the next numbers in the sequence. In this Concept you will learn how to do this, then you can figure out the next few values in this pattern.
Guidance
Once you have figured out a pattern rule it is easy to use that rule to extend the pattern. Extending a pattern involves writing the numbers that come next in the pattern according to the rule.
Find the next term in the following pattern: 3, 6, 9, 12, ____
First, notice that this is an ascending pattern meaning that it will involve addition, multiplication or both.
What is the relationship between these numbers? How were they increased?
If you think about this question, you can see that each number was increased by adding three.
To extend the pattern, we simply add three to the last number in the sequence.
\begin{align*}12 + 3 = 15\end{align*}
Our answer is 15.
Sometimes we need to extend patterns in other ways too. Remember how we used a variable to create an expression which described the pattern rule? We can use these expressions to extend number patterns. Sometimes, a table called a function table can help us organize our values as we extend the pattern. A function is a set of ordered pairs in which each element of the domain \begin{align*}(x)\end{align*} has only one element associated with it in the range \begin{align*}(y)\end{align*}.
Let’s apply this information to one of the patterns from the first section. Our pattern was 1, 3, 11, 43 and the pattern rule was \begin{align*}4x - 1\end{align*}. If \begin{align*}y\end{align*} is the next number in the pattern, then we can write the expression \begin{align*}y = 4x - 1\end{align*}.
Let’s organize these values into a table.
\begin{align*}y = 4x - 1\end{align*}
position | \begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|---|
1st | 1 | 3 |
2nd | 3 | 11 |
3rd | 11 | 43 |
4th | 43 | 171 |
5th | 171 |
Notice how \begin{align*}x\end{align*} is the number in the pattern and \begin{align*}y\end{align*} is the next number in the pattern. Theoretically, any number could be \begin{align*}x\end{align*}, but for the purposes of extending a numerical pattern, \begin{align*}x\end{align*} has to be the last number in the pattern. When the number 43 is entered into the equation as the value for \begin{align*}x\end{align*}, the value of \begin{align*}y\end{align*} is 171. 171 is the 5th number in the pattern.
Sometimes using a table can be helpful and other times it can be confusing. As long as you understand the pattern rule, you can extend the pattern by applying the rule.
Sometimes, you need to extend rule by looking far out into the future.
What is the seventh number in the sequence: 1, 3, 9, 27, = ____
First, let’s figure out the rule. This is an ascending sequence so it uses addition, multiplication or both. The rule in this case is \begin{align*}\times \ 3\end{align*}. Using the rule we can write the following expression \begin{align*}y=3x\end{align*}.
Next we can organize this information into a table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 3 |
3 | 9 |
9 | 27 |
27 | 81 |
81 | 243 |
243 | 729 |
729 | 2187 |
Our answer is 2187.
Now it's time for you to try a few on your own.
Example A
9, 17, 33, ___, ___
Solution: 65, 129
Example B
3, 10, 31, ___, ___
Solution: 94, 283
Example C
4, 17, 56, ____, ____
Solution: 173, 524
Now let's go back to the trees.
You just finished learning all about patterns. What is the rule for the Fibonacci pattern of numbers that Sara and Kelly are using?
1, 1, 2, 3, 5, 8, 13,
If you look you can see that the two previous numbers add together to equal the next number. This is the rule. Given this information, what is the next number in the pattern?
\begin{align*}8 + 13 = 21\end{align*}
What is the next one after that?
\begin{align*}13 + 21 = 34\end{align*}
By continuing to use this rule, you can continue to extend the numerical pattern.
Vocabulary
Here are the vocabulary words in this Concept.
- Pattern
- a sequence of number or geometric figures that repeats according to a pattern unit or a rule.
- Algebraic Thinking
- thinking in a mathematical way
- Numerical Patterns
- number patterns that are organized in a sequence according to a rule.
- Ascending Pattern
- a pattern that increases
- Descending Pattern
- a pattern that decreases
- Variable
- a letter used to represent an unknown quantity
- Expression
- combines variables, numbers and operations but does not equal anything because the variables can have different values.
Guided Practice
Here is one for you to try on your own.
Extend the following pattern.
24, 14, 9, ____, ____
Answer
To figure out this rule, we have to examine the operations done to each value to get to the next value.
In this example, each value was divided by 2. Then we added two to each value to get the next.
The rule is \begin{align*}y = \frac{x}{2} + 2\end{align*}.
We can determine that the next two values are 4.5 and 2.25.
This is our answer.
Video Review
Here are videos for review.
Khan Academy Patterns in Sequences 2
Khan Academy Patterns in Sequences 2
Practice
Directions: Use what you have learned to extend each numerical pattern.
1. 2, 3, 4, 5, ____, _____
2. 2, 4, 6, 8, _____, _____
3. 2, 5, 11, 23, _____,_____
4. 3, 6, 9, _____, _____
5. 16, 4, _____
6. 3, 8, 18, _____, _____
7. 100, 50, _____, _____
8. 10, 20, 30, 40, _____,_____
9. 15, 30, 45, _____,_____
10. 100, 112, 124, _____, _____
11. 12, 4, 18, 6, 21, _____
12. 40, 4, 120, 12, 130, _____
13. 2.5, _____, 7, 14, 8, 16
14. 25, 12.5, _____
15. 3, 4.5, 6, 7.5, 9, _____
Algebraic Thinking
Algebraic thinking is thinking in a mathematical way.Ascending Pattern
An ascending pattern indicates that values in the pattern are arranged from smallest to largest (to ascend means to move upward).Descending Pattern
A descending pattern indicates that values in the pattern are arranged from greatest to least (to descend means to move downward).Expression
An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Numerical Patterns
Numerical patterns are number patterns that are organized in a sequence according to a rule.Pattern
A pattern is a series of pictures, numbers or other symbols that repeat in some way according to a rule.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Image Attributions
Here you'll learn to extend numerical patterns.
Concept Nodes:
Algebraic Thinking
Algebraic thinking is thinking in a mathematical way.Ascending Pattern
An ascending pattern indicates that values in the pattern are arranged from smallest to largest (to ascend means to move upward).Descending Pattern
A descending pattern indicates that values in the pattern are arranged from greatest to least (to descend means to move downward).Expression
An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Numerical Patterns
Numerical patterns are number patterns that are organized in a sequence according to a rule.Pattern
A pattern is a series of pictures, numbers or other symbols that repeat in some way according to a rule.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.